1. Field of the Invention
This invention relates generally to a signal processing system which is specialized or programmed for (1) receiving a measured signal produced directly or indirectly from a signal that is received from a measuring device (2) to determining a mean and a covariance of a nonlinear transformation of the measured signal and (3) transmitting a resulting signal representing the determined mean and covariance in order to evoke a response related to the measured signal. The response is a physical response from a system receiving the resulting signal. This invention also relates generally to the corresponding signal processing method.
This invention also relates to a signal processing system which is programmed or specialized for estimating the expected value and covariance of a nonlinear function of a measured signal, wherein the nonlinear function is a model of a system which assumes that the measured signal is a measure of at least one of the variables of the modelled system and is a random variable, and for evoking a physical response, and to the corresponding signal processing method.
2. Discussion of the Background
The problems associated with the prior art predictions of nonlinear functions of a random variable will first be illustrated with the description of state functions of random variables and how the time evolution of the state function has been estimated.
A state function is a column vector whose components are the state variables of a system.
A state variable is one of a minimum set of numbers each of which represent the state of the system such that the set of numbers contain enough information about the history of the system to enable computation of the future behavior of the system.
A system is a set of interrelated elements whose time dependent behavior is coupled to one another.
One class of systems are physical systems. An example of a physical system is two bodies, described by their relative positions and velocities, interacting via gravitational or electromagnetic forces.
A second class of systems are engineering systems. An example of an engineering system is a combination of several pieces of equipment that are integrated with one another to perform a specific function. Examples of engineering systems are a radar tracking and/or control system, an automobile ignition system, a machine for monitoring and controlling a chemical reactions, etc.
If there were no random noise present in a system, the state function would completely determine the time evolution of the system. That is, the time evolution of the state variables (i.e., the values of the state variables at a future time), would be predetermined.
For example, the state function for the position of an object undergoing orbital motion about a large gravitational source in according with the deterministic laws of Newtonian mechanics would consist of, as the values for the state variables, the components of the position vector of the object and the velocity of the object. The state variables in this example are determined by the parametric equations of motion for the components of the position vector of the object, which in turn are derived from the Newtonian equations for the forces acting upon the object.
The time evolution of the state function may be thought of as a time transformation function acting upon the state function and which transforms the state variables from their value at a first time to their values at a second time that is subsequent to the first time. The time transformation function is usually a nonlinear function of time.
If there is random noise in the physical system, then the state variables are random variables. In this case, a time transformation function acting on the state function cannot define the time evolution of the state function or its state variables in a deterministic way. Instead, all that can be determined are expected values for the state variables and covariance between the various state variables (i.e., expected values and covariance for the state function).
The method that is most commonly used for estimating the time evolution of a physical system containing random noise is called the extended Kalman filter (EKF). The EKF method assumes, for the purpose of estimating the state function at the time increment k+1 that the estimated mean of the nonlinear state function at time increment k is approximately equal to the actual value of the nonlinear state function at the time increment k.
The state function is modelled by the set of time dependent functions which determine the state variables at a given time. At least one of the functions for the set of state variables includes a random noise term to model the noise. Essentially, the EKF method estimates the value of the state function at the time increment k+1 as a Taylor series expansion of the state function (i.e., expands the set of functions for the state variables as a Taylor series expansions) about the time increment k, and truncates the Taylor series after the first order terms.
However, values provided by the EKF method for the expected values for the state variables and the covariance for the state variables of the state function at time increment k+1 are generally not optimal. That is, given the information available at the time increment k, a more accurate estimate of the expected values and covariance of the state variables is possible. Another way to say this is that the EKF method does not propagate the state vector (i.e., the vector formed by the values of the state variables) and its covariance (i.e., the covariance between the values of the state variables) in time (i.e., predict the expected value and covariance of all the state variables at the subsequent second time based upon measured values of the state variables at the first time), as accurately as the information available at the first time about the state of the system would permit.
Moreover, the EKF method also requires calculation of a Jacobian matrix involving the first order terms, (which include first order derivatives), in the Taylor series. The first order derivatives of the Taylor series may be nonexistent, in which case the EKF method fails. In addition, the equations relating the Jacobian matrix to the covariance of the system can require an unmanageably large number of calculations. Furthermore, it is not generally possible to determine an upper limit to the uncertainties in the expected values of the state variables that are determined by the EKF method. Therefore, the EKF method is unreliable.
One example where state functions are used for tracking of an object, is in radar tracking systems where the trajectory of an object must be determined based upon measurements of its past and present position, in order to calculate its future position based upon a trajectory function of the position and velocity of the object.
Radar tracking systems send out radar signals and receive the reflected radar signals from objects. The radar signal beam moves angularly in time in order to sweep over different angular orientations. The time delay between radar signal transmission and reception is correlated to the distance of the object from the radar system. Azimuthal and/or elevational (i.e., bearing or angular) information is correlated to the time of reception of the reflected signal because of the changing direction of the radar beam. However, there are predetermined uncertainties associated with the time delay measurement and time of reception of the reflected radar signal. Therefore, there is an uncertainty associated with the value determined for the position of the object based upon the radar tracking system measurement. Thus, the determined positions of the object, that are determined by the radar tracking system, correspond to random variables. Therefore, the time transformation function for the object defining the position of the object as a function of time must be modelled as a function of random variables for the determined positions of the object, in order to account for the uncertainty in the determined position of the object.
The EKF or similar methods have been used in order to estimate the future expected position of the object and the covariance in that expected position, based upon the trajectory function modelling the trajectory of the object as a function of the positions and predetermined uncertainties in the positions of the object determined by the radar tracking system. However, this type of estimation of the future position of the object suffers from the drawbacks mentioned hereinabove in the discussion of the EKF method.
The foregoing example of a radar tracking system may also be used to illustrate a more general problem than the time evolution of a state function, which the present invention also addresses. This more general problem is the estimation of the value of a nonlinear function of a random variable of some physically significant quantity.
In the context of the example of the radar tracking system, this more general problem occurs when estimating the position and the uncertainty in the position of the object based upon the radar measurements.
There is a predetermined uncertainty in the time delay for the radar reflection signal received by the radar tracking system. Moreover, there is a predetermined uncertainty in the determination of the angular direction of the radar beam at any given time and, hence, in the determined angular orientation of the object relative to the radar tracking system. This uncertainty in the determined angular orientation occurs at least because the radar beam has an uncertainty in its intensity distribution about its cross-section that is reflected as an uncertainty in the time that the reflected signal is received. Therefore, the measurement of the time delay and the time of reception of the reflected signal are random variables.
The position and angular orientation of the object are nonlinear functions of the measured values for time delay and absolute time of radar signal reception. Since those values must be modelled as random variables to include their uncertainty, the EKF or a similar method has been used in order to determine expected values and covariance for the actual position of the object, since the actual position is a function of the random variables. Thus, the initial determination of the expected position of the object and the uncertainty associated therewith are also subject to the problems associated with the EKF method.
Most generally, a signal representing a measurement of any physical system inherently has some degree of random error associated therewith. Thus, the model of any physical system, if it is to accurately account for that random error, must include a way to estimate the expected values and uncertainties in the values of the physical system that occur due to the random error. Moreover, the quantum mechanical model for physical systems is based upon probabilities and therefore state functions of all physical systems are modelled by quantum mechanics as functions of random variables.
A signal herein is defined as any measurable quantity that is related to the changing of the physical state of a process, system, or substance. A signal includes, but is not limited to, radiation produced by a natural or a man made process, electrical fluctuations produced by a natural or a man made process, distinctive materials or chemicals produced by a natural or a man made process, distinctive structures or configurations of materials produced by a natural or a man made process, or distinctive patterns of radiation or electrical activity produced by a natural or a man made process.
The measurement of a signal is provided by a measuring device. A measuring device as defined herein may be, but is not limited to, any physical device that interacts with a physical system and provides information that is ultimately represented in the form of either a mean and a covariance or a form that can be converted into a mean and a covariance. A measuring device as defined herein includes any device that emits a signal and measures the change of the signal upon its return, and a device that measures a signal that is naturally produced by a physical process, or any device that measures a signal that is produced by a man made process.
The EKF method has been predominant in all relevant applications involving calculation of expected values and uncertainties in the expected values for nonlinear functions of random variables, including determining the time evolution of state functions of physical systems, for over two decades. During that time there has been a continuing need for more accurately and easily estimating the expected values and covariance of nonlinear functions of random variables.